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If I have an equipped creature and my opponent Mind Controls it, does my equipment remain attached to it? Can I re-equip that equipment to one of my creatures on my turn? I assume since I control the equipment I'm the only one that can play its activated abilities because it was only the equipped creature that was mind-controlled.

This answer is incorrect.The ruling refers to "control of" auras or Equipments. The Aura or Equipment is still attached to the creature, but is controlled by the other player.
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tengfredMar 7 '13 at 7:53

A tricky case is with Blazing Torch. It gives the creature an activated ability so you might think you can use it? The answer is no, this because one of the costs of activating it is sacrificing the torch itself and you can't sacrifice another player's permanents!

Another tricky equipment is Umezawa's Jitte. Note that the equipment is the one with all the abilities...

Your equipment remains attached to the Mind Controlled creature, because no rule or card effect has indicated otherwise.

Yes, you may Equip the the equipment to one of the creatures you control on your turn. The equip activated ability can normally only be activated any time you could play a sorcery, but other cards (i.e. Leonin Shikari) might modify that. At this point, it will become unattached from the former creature.

702.6a Equip is an activated ability of Equipment cards. “Equip [cost]” means “[Cost]: Attach this permanent to target creature you control. Activate this ability only any time you could cast a sorcery.”

Yes, you are the only one that can play the equipment's activated abilities, because you are the equipment's controller. Keep in mind though, if the attached equipment causes the creature to gain an activated ability, the gained ability isn't an ability of the equipment. (example: Diviner's Wand)

301.5d An Equipment’s controller is separate from the equipped creature’s controller; the two need not be the same. Changing control of the creature doesn’t change control of the Equipment, and vice versa. Only the Equipment’s controller can activate its abilities. However, if the Equipment grants an ability to the equipped creature (with “gains” or “has”), the equipped creature’s controller is the only one who can activate that ability.

112.1a An ability is a characteristic an object has that lets it affect the game. An object’s abilities are defined by its rules text or by the effect that created it. Abilities can also be granted to objects by rules or effects. (Effects that do so use the words “has,” “have,” “gains,” or “gain.”) Abilities generate effects. (See rule 609, “Effects.”)

You are correct. The equipment stays attached to the stolen creature, but you still control it and may re-equip it to a different creature that you control.

Here are the rules for equipment (emphasis added):

301.5. Some artifacts have the subtype “Equipment.” An Equipment can be attached to a creature. It can’t legally be attached to an object
that isn’t a creature.

301.5a The creature an Equipment is attached to is called the “equipped creature.” The Equipment is attached to, or “equips,” that
creature.

301.5b An Equipment is cast and enters the battlefield just like any other artifact. An Equipment doesn’t enter the battlefield attached to
a creature. The equip keyword ability attaches the Equipment to a
creature you control (see rule 702.6, “Equip”). Control of the
creature matters only when the equip ability is activated and when it
resolves. Spells and other abilities may also attach an Equipment to a
creature. If an effect attempts to attach an Equipment to an object
that can’t be equipped by it, the Equipment doesn’t move.

301.5c An Equipment that’s also a creature can’t equip a creature. An Equipment that loses the subtype “Equipment” can’t equip a creature.
An Equipment can’t equip itself. An Equipment that equips an illegal
or nonexistent permanent becomes unattached from that permanent but
remains on the battlefield. (This is a state-based action. See rule
704.) An Equipment can’t equip more than one creature. If a spell or ability would cause an Equipment to equip more than one creature, the
Equipment’s controller chooses which creature it equips.

301.5d An Equipment’s controller is separate from the equipped creature’s controller; the two need not be the same. Changing control
of the creature doesn’t change control of the Equipment, and vice
versa. Only the Equipment’s controller can activate its abilities.
However, if the Equipment grants an ability to the equipped creature
(with “gains” or “has”), the equipped creature’s controller is the
only one who can activate that ability.

And here's the exact wording of the equip ability (emphasis added):

702.6a Equip is an activated ability of Equipment cards. “Equip [cost]” means “[Cost]: Attach this permanent to target creature you
control. Activate this ability only any time you could cast a
sorcery.”

So, essentially, only the equip ability itself specifies "creature you control." If the equipment is already attached, the only restrictions are:

The equipped permanent must still be a creature (301.5).

The equipment cannot itself be a creature (301.5c).

The equipment can't be attached to a creature with protection from it, due to the definition of protection (702.15).